When the Lie group \(\mathrm {PSL}(2,\mathbb {R})\)  is replaced with a higher rank one, the relevant geometric object is not always immediately clear. In rank 2 however, combined work of Schoen [1]}, Labourie [2]}, Loftin [3]}, Collier [4]}, and Collier-Tholozan-Toulisse [5]}, provides a geometric interpretation to representations in the various distinguished components of the relevant character variety. These components are usually maximal components or Hitchin components, which maximize a topological quantity, the Toledo invariant, or contain a deformation of the classical Teichmüller space.
